Distributive extensions and quasi-framal algebras

Tah-kai Hu
1966 Canadian Journal of Mathematics - Journal Canadien de Mathematiques  
Introduction. In (2; 3; 4) , A. L. Foster denned Boolean extensions of framal algebras and bounded Boolean extensions of framal-in-the-small algebras. Foster proved that the class of Boolean (of bounded Boolean) extensions of a framal (a framal-in-the-small) algebra A is coextensive up to isomorphism with a certain class of subdirect powers of A, namely, the class of normal (of bounded normal) subdirect powers of A. His proofs apply, however, to considerably more general situations. Indeed, as
more » ... emarked in (2) , the construction of Boolean extensions may be carried out for an arbitrary universal algebra with finitary operations; this is done, in fact, in (4). Using precisely the same methods of proof as those in (2; 3; 4), we extend some of Foster's results in two directions : 1. We allow the algebras in question to admit (possibly) infinitary operations. 2. We construct extensions of algebras by using distributive lattices in place of Boolean algebras. This leads us to consider quasi-framal algebras, which include framal algebras in Foster's sense. We devote this paper to the statement of these generalizations. In order to establish them in a self-contained exposition, we shall reproduce Foster's proofs in full (and, in places, in greater detail and somewhat more precise notations). In §2, we give certain preliminaries and go to some lengths to clarify the foundations of our subject; here we introduce the concept of functional rank of a species of algebras, on which the later developments lean rather heavily. In § §3-7, we construct certain extensions of an arbitrary universal algebra. The distributive extension defined generalizes directly the Boolean extension in Foster's sense and is, as described by Foster, a kind of pseudo-hypercomplex algebra; while the lattice extension is defined by purely formal analogy with Foster's extension and is related to the distributive extensions in much the same way as the ring of formal power series over a ring R is related to the ring of polynomials over R. In §8, we prove some structure theorems-in terms of subdirect factorizations-and this is accomplished, a little surprisingly, for entirely arbitrary universal algebras, with no particular identities assumed. Special classes of algebras such as quasi-framal algebras are treated in § §9 and 10. For these algebras, stronger structure theorems may be derived. Finally, we consider framal algebras in §11, now, however, in the general setting of quasi-framal algebras.
doi:10.4153/cjm-1966-029-6 fatcat:sb2dc2sdofe4nia5y4ckjg3zii